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Hadamard's Maximum Determinant Problem

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Hadamard's maximum determinant problem asks to find the largest possible determinant (in absolute value) for any n×n matrix whose elements are taken from some set. Hadamard (1893) proved that the determinant of any complex n×n matrix A with entries in the closed unit disk |a_(ij)|<=1 satisfies

|detA|<=n^(n/2),
(1)

with equality attained by the Vandermonde matrix of the n roots of unity (Faddeev and Sominskii 1965, p. 331; Brenner and Cummings 1972). The first few values for max|detA_n| for n=1, 2, ... are 1, 2, 3sqrt(3), 16, 25sqrt(5), 216, ..., and the squares of these are 1, 4, 27, 256, 3125, ... (OEIS A000312).

A (-1,1)-matrix attaining the bound n^(n/2) is known as a Hadamard matrix (Brenner and Cummings 1972). The same bound applies to all (-1,1)-matrices, and sharper bounds are known for orders n>2 that are not a multiple of 4. A summary of what is known about such bounds is given by Orrick and Solomon.

For a (0,1)-matrix, Hadamard's bound can be improved to

|detA|<=((n+1)^((n+1)/2))/(2^n)
(2)

(Faddeev and Sominskii 1965, problem 523; Brenner and Cummings 1972).

For an n×n (0,1)-matrix (i.e., a binary matrix), the largest possible determinants beta_n for n=1, 2, ... are 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, ... (OEIS A003432). The numbers of distinct n×n binary matrices having the largest positive determinant are 1, 3, 3, 60, 3600, 529200, 75600, 195955200, 13716864000, ... (OEIS A051752).

nmatrices
1[1]
2[1 0; 0 1],[1 0; 1 1],[1 1; 0 1]
3[0 1 1; 1 0 1; 1 1 0],[1 0 1; 1 1 0; 0 1 1],[1 1 0; 0 1 1; 1 0 1]

For an n×n (-1,1)-matrix, the largest possible determinants alpha_n for n=1, 2, ... are 1, 2, 4, 16, 48, 160, ... (OEIS A003433; Ehlich and Zeller 1962, Ehlich 1964). The numbers of distinct n×n (-1,1)-matrices having the largest positive determinant are 1, 4, 96, 384, 30720, ... (OEIS A188895, indexed there by n-1). alpha_n is related to the largest possible (0,1)-matrix determinant beta_(n-1) by

alpha_n=2^(n-1)beta_(n-1)
(3)

for n>=2 (Williamson 1946, Brenner and Cummings 1972).

nmatrices
1[1]
2[-1 -1; 1 -1],[-1 1; -1 -1],[1 -1; 1 1],[1 1; -1 1]

For an n×n (-1,0,1)-matrix, the largest possible determinants gamma_n are the same as alpha_n (Ehlich 1964, Brenner 1972). The numbers of n×n (-1,0,1)-matrices having maximum positive determinants are 1, 4, 240, 384, 30720, ... (OEIS A051753).

For order 198, Georgiou and Stylianou (2026) attained the Ehlich-Wojtas bound by constructing a D-optimal matrix. Thus the largest absolute determinant of a (-1,1)-matrix of that order is 2^(99)·197·98^(98).

See also

(0,1)-Matrix, (-1,0,1)-Matrix, (-1,1)-Matrix, Determinant, Hadamard Matrix, Integer Matrix

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References

Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Cohn, J. H. E. "Determinants with Elements +/-1." J. London Math. Soc. 14, 581-588, 1963.Ehlich, H. "Determinantenabschätzungen für binäre Matrizen." Math. Z. 83, 123-132, 1964.Ehlich, H. and Zeller, K. "Binäre Matrizen." Z. angew. Math. Mechanik 42, T20-21, 1962.Faddeev, D. K. and Sominskii, I. S. Problems in Higher Algebra. San Francisco, CA: W. H. Freeman, 1965.Georgiou, S. D. and Stylianou, S. "An Exact Compression-And-Lifting Procedure and D-Optimal Designs of Orders 198 and 170." 24 Oct 2026. https://arxiv.org/abs/2610.33418.Hadamard, J. "Résolution d'une question relative aux déterminants." Bull. Sci. Math. 17, 240-246, 1893.Hall, M. Combinatorial Theory, 2nd ed. New York: Wiley, 1998.Kaplansky, I. "Never Too Late." Amer. Math. Monthly 102, 259, 1995.MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland, p. 54, 1978.Orrick, W. and Solomon, B. "Known Bounds on Maximal Determinants." https://web.archive.org/web/20120930030439/http://www.indiana.edu/%7Emaxdet/bounds.html.Sloane, N. J. A. Sequences A000312, A003432/M0720, A003433/M1291, A051752, A051753, and A188895 in "The On-Line Encyclopedia of Integer Sequences."Williamson, J. "Determinants Whose Elements Are 0 and 1." Amer. Math. Monthly 53, 427-434, 1946.Yang, C. H. "Some Designs for Maximal (+1,-1)-Determinant of Order n=2 (mod 4)." Math. Comput. 20, 147-148, 1966.Yang, C. H. "A Construction for Maximal (+1,-1)-Matrix of Order 54." Bull. Amer. Math. Soc. 72, 293, 1966.Yang, C. H. "On Designs of Maximal (+1,-1)-Matrices of Order n=2 (mod 4)." Math. Comput. 22, 174-180, 1968.Yang, C. H. "On Designs of Maximal (+1,-1)-Matrices of Order n=2 (mod 4) II." Math. Comput. 23, 201-205, 1969.

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Hadamard's Maximum Determinant Problem

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Weisstein, Eric W. "Hadamard's Maximum Determinant Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HadamardsMaximumDeterminantProblem.html

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